![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > znegcl | Structured version Visualization version GIF version |
Description: Closure law for negative integers. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
znegcl | ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elz 11580 | . 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
2 | negeq 10474 | . . . . . 6 ⊢ (𝑁 = 0 → -𝑁 = -0) | |
3 | neg0 10528 | . . . . . 6 ⊢ -0 = 0 | |
4 | 2, 3 | syl6eq 2820 | . . . . 5 ⊢ (𝑁 = 0 → -𝑁 = 0) |
5 | 0z 11589 | . . . . 5 ⊢ 0 ∈ ℤ | |
6 | 4, 5 | syl6eqel 2857 | . . . 4 ⊢ (𝑁 = 0 → -𝑁 ∈ ℤ) |
7 | nnnegz 11581 | . . . 4 ⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ) | |
8 | nnz 11600 | . . . 4 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℤ) | |
9 | 6, 7, 8 | 3jaoi 1538 | . . 3 ⊢ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) → -𝑁 ∈ ℤ) |
10 | 9 | adantl 467 | . 2 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) → -𝑁 ∈ ℤ) |
11 | 1, 10 | sylbi 207 | 1 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∨ w3o 1069 = wceq 1630 ∈ wcel 2144 ℝcr 10136 0cc0 10137 -cneg 10468 ℕcn 11221 ℤcz 11578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-nn 11222 df-z 11579 |
This theorem is referenced by: znegclb 11615 nn0negz 11616 zsubcl 11620 zeo 11664 zindd 11679 znegcld 11685 zriotaneg 11692 uzneg 11906 zmax 11987 rebtwnz 11989 qnegcl 12007 fzsubel 12583 fzosubel 12734 ceilid 12857 modcyc2 12913 expsub 13114 seqshft 14032 climshft 14514 znnenlem 15145 negdvdsb 15206 dvdsnegb 15207 summodnegmod 15220 dvdssub 15234 odd2np1 15272 divalglem6 15328 bitscmp 15367 gcdneg 15450 neggcd 15451 gcdaddmlem 15452 gcdabs 15457 lcmneg 15523 neglcm 15524 lcmabs 15525 mulgaddcomlem 17770 mulgneg2 17782 mulgsubdir 17789 cycsubgcl 17827 zaddablx 18481 cyggeninv 18491 zsubrg 20013 zringmulg 20040 zringinvg 20049 aaliou3lem9 24324 sinperlem 24452 wilthlem3 25016 basellem3 25029 basellem4 25030 basellem8 25034 basellem9 25035 lgsneg 25266 lgsdir2lem4 25273 lgsdir2lem5 25274 ex-fl 27640 ex-mod 27642 pell1234qrdich 37944 rmxyneg 38004 monotoddzzfi 38026 monotoddzz 38027 oddcomabszz 38028 jm2.24 38049 acongtr 38064 fzneg 38068 jm2.26a 38086 cosknegpi 40592 enege 42076 onego 42077 0nodd 42328 2zrngagrp 42461 zlmodzxzequap 42806 flsubz 42830 digvalnn0 42911 dig0 42918 dig2nn0 42923 |
Copyright terms: Public domain | W3C validator |